Killing vector fields and a homogeneous isotropic universe
V.A. Steklov Mathematical Institute, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991, Russian Federation
Some basic theorems on Killing vector fields are reviewed. In particular, the topic of a constant curvature space is examined. A detailed proof is given for a theorem describing the most general form of the metric of a homogeneous isotropic space-time. Although this theorem can be considered commonly known, its complete proof is difficult to find in the literature. An example metric is presented which, while all its spatial cross sections correspond to constant curvature spaces, still is not homogeneous and isotropic as a whole. An equivalent definition of a homogeneous isotropic space-time in terms of embedded manifolds is also given.